An explicit self-dual construction of complete cotorsion pairs in the relative context

  • Leonid Positselski

    Czech Academy of Sciences, Prague, Czechia
An explicit self-dual construction of complete cotorsion pairs in the relative context cover
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Abstract

Let be a homomorphism of associative rings, and let be a hereditary complete cotorsion pair in . Let be the cotorsion pair in in which is the class of all left -modules whose underlying -modules belong to . Assuming that the -resolution dimension of every left -module is finite and the class is preserved by the coinduction functor , we show that is the class of all direct summands of left -modules finitely (co)filtered by -modules coinduced from -modules from . Assuming that the class is closed under countable products and preserved by the functor , we prove that is the class of all direct summands of left -modules cofiltered by -modules coinduced from -modules from , with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from have finite -resolution dimension bounded by , involves cofiltrations indexed by the ordinal . The dual results also hold, provable by the same technique going back to the author's monograph on semi-infinite homological algebra~(2010). In addition, we discuss the -cotilting and -tilting cotorsion pairs, for which we obtain better results using a suitable version of a classical Bongartz–Ringel lemma. As an illustration of the main results of the paper, we consider certain cotorsion pairs related to the contraderived and coderived categories of curved DG-modules.

Cite this article

Leonid Positselski, An explicit self-dual construction of complete cotorsion pairs in the relative context. Rend. Sem. Mat. Univ. Padova 149 (2023), pp. 191–253

DOI 10.4171/RSMUP/118